If   h(x)=(x−x2)(2x2−x3)   then   h′(x)  is -------------
 

To find the derivative \(h'(x)\) of the function \(h(x) = (x - x^2)(2x^2 - x^3)\), we can apply the **product rule**. The product rule states that if \(h(x) = f(x) \cdot g(x)\), then:

\[
h'(x) = f'(x)g(x) + f(x)g'(x).
\]

In this case, let's define:
\[
f(x) = x - x^2, \quad g(x) = 2x^2 - x^3.
\]

### Step 1: Find \(f'(x)\) and \(g'(x)\)

1. **Compute \(f'(x)\):**
\[
f(x) = x - x^2 \implies f'(x) = 1 - 2x.
\]

2. **Compute \(g'(x)\):**
\[
g(x) = 2x^2 - x^3 \implies g'(x) = 4x - 3x^2.
\]

### Step 2: Apply the product rule

Using the product rule, we have:
\[
h'(x) = f'(x)g(x) + f(x)g'(x).
\]

Substitute \(f(x), f'(x), g(x),\) and \(g'(x)\) into the equation:

\[
h'(x) = (1 - 2x)(2x^2 - x^3) + (x - x^2)(4x - 3x^2).
\]

### Step 3: Simplify the expression

Let's expand each term:

1. **Expand \(f'(x)g(x)\):**
\[
(1 - 2x)(2x^2 - x^3) = (1)(2x^2 - x^3) + (-2x)(2x^2 - x^3).
\]
\[
= 2x^2 - x^3 - 4x^3 + 2x^4.
\]
\[
= 2x^2 - 5x^3 + 2x^4.
\]

2. **Expand \(f(x)g'(x)\):**
\[
(x - x^2)(4x - 3x^2) = (x)(4x - 3x^2) + (-x^2)(4x - 3x^2).
\]
\[
= 4x^2 - 3x^3 - 4x^3 + 3x^4.
\]
\[
= 4x^2 - 7x^3 + 3x^4.
\]

### Step 4: Combine like terms

Now combine both expressions:

\[
h'(x) = (2x^2 - 5x^3 + 2x^4) + (4x^2 - 7x^3 + 3x^4).
\]

Combine like terms:

\[
h'(x) = (2x^2 + 4x^2) + (-5x^3 - 7x^3) + (2x^4 + 3x^4).
\]
\[
h'(x) = 6x^2 - 12x^3 + 5x^4.
\]

### Final Answer

\[
\boxed{h'(x) = 5x^4 - 12x^3 + 6x^2.}
\]

Would you like more details on this problem or have any questions?

Here are 5 related questions for further exploration:
1. How would you find the second derivative \(h''(x)\) of the function \(h(x)\)?
2. What are the critical points of the function \(h(x)\)?
3. Can you determine the intervals where \(h(x)\) is increasing or decreasing?
4. How can you find the inflection points of \(h(x)\)?
5. What would the graph of \(h(x)\) look like, and where would it have local maxima or minima?

**Tip:** When dealing with derivatives of products of functions, always consider using the product rule for a structured approach.

Learn how to find the derivative of the function h(x) = (x - x^2)(2x^2 - x^3) using the product rule. This problem involves polynomial functions and derivative rules, suitable for advanced high school students (Grades 11-12).

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